Mathematics, Statistics, and Physics
https://soar.wichita.edu/handle/10057/119
2021-10-24T06:52:44ZTorus actions, maximality, and non-negative curvature
https://soar.wichita.edu/handle/10057/22214
Torus actions, maximality, and non-negative curvature
Escher, Christine; Searle, Catherine
Let $ℳ_0^n$ be the class of closed, simply connected, non-negatively curved Riemannian n-manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if $M \in ℳ_0^n$ then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all $M \in ℳ_0^n$. Finally, we show the Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action.
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2021-09-03T00:00:00ZApplied topological data analysis to El-Niño Southern Oscillation
https://soar.wichita.edu/handle/10057/21758
Applied topological data analysis to El-Niño Southern Oscillation
Shirazbakhti, Ghazaleh
The El Niño Southern Oscillation (ENSO) is one of the most powerful climate phenomena
that can change global air circulation, affecting temperature and rainfall around the planet. ENSO
has three phases: El Niño and La Niña are two extreme phases of the ENSO cycle, and ENSO
neutral is a transitional period between El Niño and La Niña.
Topological data analysis (TDA) is an innovative approach which focuses on a data set's
"shape" or topological structures such loops, holes, and voids. We used TDA to describe the
homology groups of the two-dimensional function determined by sea surface temperatures of
tropical Pacific Ocean. The persistent homology of the function, which is determined by the
monthly mean sea surface temperature in an expressed area of the Pacific, was used to identify
ENSO phases. We classify using TDA summaries and compare the results to the existing NOAA
classification utilizing topology information of sea surface temperatures of correctly specified
ENSO phases.
We compute the morse filtration's persistent homology for each month, then utilize the
ECC to summarize the persistent homology. To reduce arbitrary errors and make the curve
continuous, we utilized the smoothing function. We used LDA algorithm as a classification
method and for dimension reduction, we used PCA transformation.
Thesis (M.S.)-- Wichita State University, College of Liberal Arts and Sciences, Dept. of Mathematics, Statistics, and Physics
2021-07-01T00:00:00ZRicci tensor under conformal change of metric as an elementary obstruction to certain Einstein metrics
https://soar.wichita.edu/handle/10057/21757
Ricci tensor under conformal change of metric as an elementary obstruction to certain Einstein metrics
Shabo, Faiz
Gregorio Ricci-Curbastro, (1853-1925) Bologna, was an Italian mathematician and
a professor at the University of Padua from 1880 -1925. He was the rst to introduce the
systematic theory of tensor analysis in 1887 with a a major contribution later by his student
Tullio Levi-Civita. However, the roots of tensor analysis were laid by the work of German
mathematician Bernhard Riemann in Di erential Geometry. The beginning of the twentieth
century was the emergence of the study of the Ricci Tensor due to its major role in the
mathematical formulation of the theory of general relativity of Albert Einstein. Also, as one
of the important geometric features that have been used to prove several major theorems in
di erential geometry and topology. Here I will focus on the idea of the conformal change of
metrics and what is the necessary and su cient condition for a metric to be conformal to
another metric on the same manifold, and how geometric objects like Riemannian and Ricci
curvature are related under this change.
Thesis (M.S.)-- Wichita State University, College of Liberal Arts and Sciences, Dept. of Mathematics, Statistics, and Physics
2021-07-01T00:00:00ZPositive (p, n)-intermediate scalar curvature and Gromov-Lawson cobordism
https://soar.wichita.edu/handle/10057/21735
Positive (p, n)-intermediate scalar curvature and Gromov-Lawson cobordism
Burkemper, Matthew Bryan
In this thesis we consider a well known construction due to Gromov, Lawson, and Gajer
which allows for the extension of a metric of positive scalar curvature over the trace of a
surgery in codimension at least 3 to a metric of positive scalar curvature which is a product
near the boundary. We generalize this construction to work for (p; n)-intermediate scalar
curvature for $0 \leq p \leq n-2$ for surgeries in codimension at least p + 3.
Thesis (Ph.D.)-- Wichita State University, College of Liberal Arts and Sciences, Dept. of Mathematics, Statistics, and Physics
2021-08-01T00:00:00Z